Geodesic
Generating elements for groups and Lie algebras of the form $F/[N,N]$
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 2, pp. 60-71 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $F$ be a free product of groups $A_i~(i\in I)$ and a free group $G$ and its normal subgroup $N$ has trivial intersection with each factor $A_i$. Subject to these conditions we will establish necessary and sufficient conditions for an element of the group $F/[N,N]$ belongs to the subgroup generated by a given finite set of elements of $F/[N,N]$ and necessary and sufficient conditions for a given set of elements of the group $F/[N,N]$ to generate it. Similar results are proved also for Lie algebras.
Keywords: group ring, Lie algebra, universal enveloping algebra. 
@article{VNGU_2015_15_2_a4,
     author = {A. F. Krasnikov},
     title = {Generating elements for groups and {Lie} algebras of the form $F/[N,N]$},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {60--71},
     year = {2015},
     volume = {15},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/}
}
TY  - JOUR
AU  - A. F. Krasnikov
TI  - Generating elements for groups and Lie algebras of the form $F/[N,N]$
JO  - Sibirskij žurnal čistoj i prikladnoj matematiki
PY  - 2015
SP  - 60
EP  - 71
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/
LA  - ru
ID  - VNGU_2015_15_2_a4
ER  - 
%0 Journal Article
%A A. F. Krasnikov
%T Generating elements for groups and Lie algebras of the form $F/[N,N]$
%J Sibirskij žurnal čistoj i prikladnoj matematiki
%D 2015
%P 60-71
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/
%G ru
%F VNGU_2015_15_2_a4
A. F. Krasnikov. Generating elements for groups and Lie algebras of the form $F/[N,N]$. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 2, pp. 60-71. http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/

[1] Birman J. S., “An Inverse Function Theorem for Free Groups”, Proc. Am. Math. Soc., 41:2 (1973), 634–638 | DOI | MR | Zbl

[2] A. F. Krasnikov, “Generators of the group $F/[N,N]$”, Math. Notes, 24:2 (1978), 591–594 | DOI | MR

[3] C. K. Gupta, E. I. Timoshenko, “Generating elements for groups of the form $F/R^\prime$”, Algebra Logic, 40:3 (2001), 137–143 | DOI | MR | Zbl

[4] N. S. Romanovskii, “Shmel'kin embeddings for abstract and profinite groups”, Algebra Logic, 38:5 (1999), 326–334 | DOI | MR | Zbl

[5] U. U. Umirbaev, “Partial derivatives and endomorphisms of some relatively free Lie algebras”, Sib. Math. J., 34:6 (1993), 1161–1170 | DOI | MR | Zbl

[6] V. V. Talapov, “Relative linear dependence problem for the variety $\mathfrak{A}\mathfrak{N}_c$ of Lie algebras”, Sib. Math. J., 25:6 (1984), 979–985 | DOI | MR | Zbl

[7] N. S. Romanovskii, “The occurrence problem for extensions of Abelian groups by nilpotent groups”, Sib. Math. J., 21:2 (1980), 273–276 | DOI | MR

[8] V. N. Remeslennikov, “Example of a finitely presented group in the variety $\mathfrak{A}^5$ with the unsolvable word problem”, Algebra Logic, 12:5 (1973), 327–346 | DOI | MR | Zbl

[9] A. L. Shmel'kin, A. V. Syrtsov, “On embeddings of some quotient algebras of free sums of Lie algebras”, J. Math. Sci., New York, 140:2 (2007), 340–344 | DOI | MR | Zbl

[10] N. Jacobson, Lie Algebras, John Wiley Sons, New York–London, 1962 | MR | MR | Zbl